Question

In figure, AT is a tangent to the circle with centre O such that OT = 4 cm
and ZOTA = 30°. Find the length of the tangent AT.
O.
4lon​

Answer 1

Given :-

• A circle with centre 0

such that

• AT is a tangent to a circle from external point T.

• OT = 4 cm

• ∠OTA = 30°

To Find :-

• Length of tangent AT.

Solution :-

Since,

• AT is tangent to a circle

and

• OA is radius of circle.

As we know that,

• Radius and tangent are perpendicular.

So,

• OA is perpendicular to AT.

Now,

$\rm :\longmapsto\:In \: right \: \triangle \: OAT$

$\rm :\longmapsto\: cos30 \degree\: = \: \dfrac{TA}{OT}$

$\rm :\longmapsto\:\dfrac{ \sqrt{3} }{2} = \dfrac{TA}{4}$

$\bf\implies \:TA \: = \: 2 \sqrt{3} \: cm$

Additional Information :-

• The tangent line never crosses the circle, it just touches the circle.

• At the point of tangency, it is perpendicular to the radius.

• A chord and tangent form an angle and this angle is same as that of tangent inscribed on the opposite side of the chord.

• From the same external point, the tangent segments to a circle are equal.

• Tangent are equally inclined to the line segment joining centre and external point.